Elementary matrix transformations

Question

Prove that pre-multiplying a matrix $A_m $ by the elementary matrix obtained with any matrix elementary line transformation $ I_m \underset{l_1 \leftrightarrow l_2}{\longrightarrow} E $ is the same as applying said elementary line transformation on the matrix $ A_m $.

Answer 1

Let $e_1,\dots, e_n$ be the standard basis row vectors, and observe the following facts about matrix multiplication:

1. $e_iA$ gives the $i$th row of the matrix $A$
2. For row vectors $v_1,\dots, v_m$, we have $$\pmatrix{\ v_1\ \\ \vdots \\ v_m} A=\pmatrix{\ v_1A\ \\ \vdots \\ v_mA} $$

Combine these to get the desired result for any elementary row operations.